Advertisement

Diffusive Instabilities and Pattern Formation in the Belousov-Zhabotinsky System

  • A. B. Rovinsky
Chapter
  • 106 Downloads
Part of the NATO ASI Series book series (NSSB, volume 244)

Abstract

Complex dynamics of certain types of chemical systems like the Belousov-Zhabotinsky reaction has been attracting the attention of many researchers for nearly three decades [1, 3, 16]. This dynamics is governed by underlying nonlinear kinetics and manifests itself as multistability and periodiC., multiperiodic or chaotic oscillations in perfectly stirred systems. At the same time, the investigations of spatial systems have mainly been confined to consideration of classical kinds of waves, or “autowaves” [1, 4]. This term means that a local event, e.g. a pulse generated by a cell, is transferred by diffusion to neighbouring points. Such transport triggers the similar events there, and thus gives rise to a travelling wave. Diffusion plays rather a passive role here and dynamics of the whole medium is determined by dynamic properties of its local elements. Rashevsky [9] and Turing [15] found, however, that diffusion may be as important for a system’s dynamics as its local kinetics. That means that behaviour of a spatial system may be far more complex than one can conceive from consideration of its local characteristics. Turing [15] was the first who clearly described the possible destabilizing role of diffusion.

Keywords

Hopf Bifurcation Spatial System Turing Instability Homogeneous Steady State Diffusive Instability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Field, R.J. & Burger, M. (eds.) (1985). Oscillations and Traveling Waves in Chemical Systems. Wiley Interscience: New York.Google Scholar
  2. [2]
    Grassberger, P. & Procaccia, I. (1983). Measuring the Strangeness of strange attractors. Physica D9, 189.zbMATHMathSciNetGoogle Scholar
  3. [3]
    J. Phys. Chem. (1989), 93, 7. The issue dedicated to Prof. Noyés’ 70th birthday.Google Scholar
  4. [4]
    Krinsky, V.I. (1984). Self-organization. Autowaves and Structures far from Equilibrium. Springer: Berlin.CrossRefzbMATHGoogle Scholar
  5. [5]
    Kuhnert, L. (1983). Chemishe Structur Bildung in Festen Gelen auf der Basis der Belousov-Zhabotinsky reaction. Naturwis. 70, 464.ADSCrossRefGoogle Scholar
  6. [6]
    Kuhnert, L., Yamaguchi, T., Nagy-Ungvarai, Z. , Müller, S.C. & Hess, B. (1989). Wave Propagation and Pattern Formation in Catalyst-Immobilized Gels. In International Conference on Dynamics of Exotic Phenomena in Chemistry, Preprints of Lectures and Abstracts of Posters, p. 289. Hajduszobozlo: Hungary.Google Scholar
  7. [7]
    Kuramoto, Y. (1984). Turbulence and Waves. Springer: Berlin.zbMATHGoogle Scholar
  8. [8]
    Maselko, J., Reckley, J.S. & Showalter, K. (1989). Regular and Irregular Spatial Patterns in an Immobilized-Catalyst Belousov-Zhabotinsky Reaction. J. Phys. Chem. 93, 2774.CrossRefGoogle Scholar
  9. [9]
    Rashevskky, N. (1940). An Approach to the Mathematical Biophysics of Biological Self-Regulation and of Cell Polarity. Bull. Math. Biophys. 2, 15.CrossRefGoogle Scholar
  10. [10]
    Rovinsky, A.B. & Zhabotinsky, A.M. (1984). Mechanism and Mathematical Model of the Oscillating Bromate-Ferroin- Bromomalonic Acid Reaction. J. Phys. Chem. 88, 6084.CrossRefGoogle Scholar
  11. [11]
    Rovinsky, A.B. (1987). Turing Bifurcation and Stationary Patterns in the Ferroin-Catalyzed Belousov-Zhabotinsky Reaction. J. Phys. Chem. 91, 4606.CrossRefGoogle Scholar
  12. [12]
    Rovinsky, A.B. (1987). Twinkling Patterns and Diffusion Induced Chaos in a Model of the Belousov-Zhabotinsky Chemical Medium. J. Phys. Chem. 91, 5113.CrossRefGoogle Scholar
  13. [13]
    Rovinsky, A.B. (1989). Stationary Patterns in a Discrete Belousov-Zhabotinsky Medium with Small Catalyst Diffusibility. J. Phys. Chem. 93, 2716.CrossRefGoogle Scholar
  14. [14]
    Takens, F. (1981). Detecting Strange Attractors in Turbulence. In Lecture Notes in Mathematics, 898, 366. Springer: New York.Google Scholar
  15. [15]
    Turing, A. (1952). The Chemical Basis of. Morphogenesis. Phil. Trans. R. Soc. 237B, 37.CrossRefGoogle Scholar
  16. [16]
    Zhabotinsky, A.M. (1974). Concentration Autooscillations. Nauka Publishing: Moscow (in Russian).Google Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • A. B. Rovinsky
    • 1
  1. 1.Institute of Biological Physics, USSR Academy of SciencePushchino Moscow RegionUSSR

Personalised recommendations